There is a non-productive Skyscraper in r35c1 for <7>. Most extension techniques are based on extending the <7> candidate. However, this Skyscraper can easily be extended into a single-step solution using a candidate other than <7>.

3a) Look for a strong link on any candidate common to (2a) and (2b); i.e., {5,9}
3b) in any cell that sees both of the endpoint cells
3c) and has a strong link to at least one of endpoint cells
3d) note: there are only six cells to check

Yes ... and No. You are using a forcing chain based on the endpoints of the Skyscraper. I like forcing chains (the Yes), but most people frown on them (the No). So, I do my best to present information as AICs.

Unfortunately, my PM has more strong links in <5> than is necessary for the general approach that I'm demonstrating. You used them to demonstrate a different approach for this PM. Had there been other cells containing <5> in [r46789c4] and [r46789c6], then my approach would have continued to work and your shortcut would have failed.

you know, I think you have a good point about the general structure. and I would be so bold as to call it an inverted w-wing.

w-wing
(x=y) - y = y - (y=x) and the x's are pincers if they both see any other x's
notice in the w-wing that the two bi-value cells have strong inferences between the candidates.

your inverted w-wing

x = (x-y) = y - y = (y-x) = x

notice how its inverted. every step of this new chain is exactly opposite a normal w-wing.
all the strong inferences are now weak and all the weak are now strong.
the cells that would contain the two bi-value cells in a w-wing now contain any number of candidates and the interaction is weak.
the candidate that is normally linked strongly in a w-wing is now linked weakly.

completely inverted to a w-wing.

Code:

w-wing: (x=y) - y = y - (y=x)
| | |
strong strong strong

inverted: x = (x-y) = y - y = (y-x) = x
| | |
weak weak weak

the x's on the ends still act like pincers obviously needing to be extensions of the weakly linked xy cells.

This appears to be a valid AIC. You assume that [r1c4] is not <5>, and then demonstrated that it led to a contradiction to the assumption. This then forces [r1c4]=5.

Just Between You and Me: I recently came to the conclusion that an AIC can be viewed as a forcing chain. I'm probably the only person who feels this way. It's all a matter of perspective. I don't expect anyone to agree with me, but you have a friend when it comes to believing in forcing chains.

[Edit: corrected spelling error.]

Last edited by daj95376 on Wed Jul 15, 2009 5:15 am; edited 1 time in total

you know, I think you have a good point about the general structure. and I would be so bold as to call it an inverted w-wing.

w-wing
(x=y) - y = y - (y=x) and the x's are pincers if they both see any other x's
notice in the w-wing that the two bi-value cells have strong inferences between the candidates.

your inverted w-wing

x = (x-y) = y - y = (y-x) = x

notice how its inverted. every step of this new chain is exactly opposite a normal w-wing.
all the strong inferences are now weak and all the weak are now strong.
the cells that would contain the two bi-value cells in a w-wing now contain any number of candidates and the interaction is weak.
the candidate that is normally linked strongly in a w-wing is now linked weakly.

completely inverted to a w-wing.

Code:

w-wing: (x=y) - y = y - (y=x)
| | |
strong strong strong

inverted: x = (x-y) = y - y = (y-x) = x
| | |
weak weak weak

the x's on the ends still act like pincers obviously needing to be extensions of the weakly linked xy cells.

Wow ... impressive!!!

You certainly did a much better job of analyzing the structure than I did. I had some concept of a hybrid cross between a W-Wing and a gM-Wing; but your perspective is much cleaner and clearer.

The thing that struck home for me was the weak link in the xy cells because they could now contain other candidates. I'm glad to see that you picked up on this as well.

so the steps in the middle could be a kite, skyscraper, a 6 link classic swordrish, etc. the coloring in the middle ciould be any shape as long as it has an even link count.

No, I hadn't tried to generalize it. I'm aware that any pattern based on an internal X-Chain can be extended. However, I wanted to keep the pattern simple enough for manual solvers to feel comfortable with it. I liked the idea of having most of the action restricted to a chute -- band/stack -- and so the Skyscraper pattern suited me perfectly.

here the 4 in r6c4 sees the 4 in r6c7.
but,
the strong link on 4's in r6c4 and r5c5 excites the skyscraper on 7's which ends in a buddy cell of the initial 4 and contains a 4. this eliminates the 4 in the ending cell.

mildly difficult to explain.
---
just as a note to anyone who uses Andrew Stuart's solver... his solver denotes this type of chain as a "AIC RULE 2" where one end of the chain starts with a candidate and ends on a difference candidate in a buddy cell containing the original candidate.

AIC RULE 1 being a chain in which the end candidates act like pincers.

Help me.
Is a closed chain with an inconsistancy called Forcing?
Is a chain ending with pinchers called AIC?
Thanks for the help and patience with me.

Easy Dan,

You're overthinking it here. A forcing chain is defined by Jeff here. He even did a great job of updating the head message in the thread to include information from discussions in the thread. However, I needed to read it numerous times to understand most of it. Here's the key part about forcing chains.

Implication Stream - a sequence of nodes and links where strong or weak inferences are made from one node to the other(s) unidirectionally from left to right. (Refer definitions for "node", "link", "strong inference" and "weak inferences" below)

Forcing Chain - a chain that has 2 or more implication streams that start from one node and end in another node where the outcomes of inferences merge from the 2 implication streams. In a forcing chain, a node can only infer the next successive node downstream.

A forcing chain starts with 2 (or more) assumptions that exhaustively cover all possibilities for some condition. When streams from each assumption result in a common conclusion, then that conclusion must be correct. Here's your streams from above. Your (exhaustive) assumptions are based on the strong link between the endpoints of the Skyscraper.

Code:

r5c4=7 => r1c4=5
r3c6=7 => r1c4=5 q.e.d.

Now, AICs are based on an inference chain that is bidirectional. You can find Myth Jellies description here. How it all works as one chain is nifty but difficult to explain w/o delving into the rules of logic. It wasn't until I learned Eureka notation that I had a comfortable feel for AIC. Even then, Asellus needed to convince me that it was okay for them to start and end with weak inferences as well.

Bottom Line (for me): A forcing chain contains multiple streams that start from a common constraint and result in a common conclusion. An AIC is a single inference chain where the bidirectional property makes a conclusion possible.

As for pincers, I hadn't heard of them until I joined this forum. For me, they add a colorful description to what's happening based on the endpoints of an AIC, but nothing more.

As for being useful, give it a try against Set XY_03 Puzzle 026. This puzzle was selected because it didn't seem to have any workarounds for the XY-Chains. Then I checked it for this pattern. It cracks the puzzle!

I agree that all of the "f" cells OR all of the "t" cells must be <5>.
The diagram has been edited to add * to two cells. One or both of these "t" cells are false (not 5) (from the skyscraper on 7). It is clear that the cells labeled "f" must be true!

I think the extended skyscraper is pretty cool. The following is more about terminology than the technique under discussion.

arkietech wrote:

Is a closed chain with an inconsistancy called Forcing?

I'm also trying to understand chain terminology. What do you think of this?

If a chain is closed it becomes a nice loop. An inconsistency is referred to as a discontinuity. If a chain closes on an inconsistency, it is a discontinuous nice loop. In the chain you wrote above,
(5)r1c4=(5-7)r3c6=(7)r3c1-(7)r5c1=(7-5)r5c4=(5)r1c4=>r1c4=5,
your loop is closed with two strong links on 5, so it's a discontinuous nice loop and the digit 5 can be placed at the discontinuity.

daj97536 wrote:

You assume that [r1c4] is not <5>, and then demonstrated that it led to a contradiction to the assumption. This then forces [r1c4]=5.

My questions are very general. I've wondered if an AIC or nice loop or any use of an inference actually requires making any assumptions whatsoever. An inference simply exists, right?. The relationship between two candidates or nodes either meet the criteria of a strong link or a weak link, or they don't. arkietech's chain starts with notation which means "The 5 in r1c4 and the 5 in r3c6 can't both be false." What assumption has been made?

I'm also trying to understand chain terminology. What do you think of this?

If a chain is closed it becomes a nice loop. An inconsistency is referred to as a discontinuity. If a chain closes on an inconsistency, it is a discontinuous nice loop. In the chain you wrote above,
(5)r1c4=(5-7)r3c6=(7)r3c1-(7)r5c1=(7-5)r5c4=(5)r1c4=>r1c4=5,
your loop is closed with two strong links on 5, so it's a discontinuous nice loop and the digit 5 can be placed at the discontinuity.

I wasn't sure what inconsistency meant to Dan, but I suspect that your use of discontinuous probably fits. For sure, a closed chain is a property of a nice/AIC loop and not a forcing chain. Sudopedia covers chains extensively, including continuous and discontinuous, so I'm not going to repeat it here.

Luke451 wrote:

daj97536 wrote:

You assume that [r1c4] is not <5>, and then demonstrated that it led to a contradiction to the assumption. This then forces [r1c4]=5.

My questions are very general. I've wondered if an AIC or nice loop or any use of an inference actually requires making any assumptions whatsoever. An inference simply exists, right?. The relationship between two candidates or nodes either meet the criteria of a strong link or a weak link, or they don't. arkietech's chain starts with notation which means "The 5 in r1c4 and the 5 in r3c6 can't both be false." What assumption has been made?

In Sudoku, an inference is a statement concerning the interaction between premises, where a premise is a statement concerning the state of the Sudoku that must be either true or false. The most common type of premise is that a particular cell has a particular candidate value. When they are used in chains or loops, the term inference is equivalent to link.

There is some discussion about the difference between inferences and implications.

There are 2 types of inference. Strong and weak:

Two premises can be linked by a strong inference if they cannot both be false.

Two premises can be linked by a weak inference if they cannot both be true.